The expected value is what you get if you add up all the products of the possible numbers of students running for president and their corresponding probabilities.
B
b
To calculate the relative frequency of an event, divide the number of times the event occurred by the total number of trials.
C
c
Make a bar graph using the table you made in Part B. Each bar should represent the number of students running for president, and the bar's height should be proportional to the corresponding relative frequency.
A
a
The expected value of students running for president of the class is 4.2. This means that on average around 4 students will run for president.
B
b
Example Table:
Number of Students Running
Frequency
Relative Frequency
1
2
2/50=0.04
2
4
4/50=0.08
3
6
6/50=0.12
4
4
4/50=0.08
5
25
25/50=0.50
6
9
23/50=0.18
C
c
Example Graph:
Practice makes perfect
a
We are asked to find the expected number of students who will run for class president. Let's look at the given graph.
For convenience, we will move the data from the graph to a table.
Number of Students Running, X
Probability, P(X)
1
0.05
2
0.15
3
0.1
4
0.15
5
0.35
6
0.2
The expected value of a discrete random variable X is the weighted sum of its values. In our case, the values of the random variable X represent the possible numbers of students running for president.
E(X)=∑ [ X* P(X)]
In other words, the expected value is what you get if you add up all the products of the possible numbers of students and their corresponding probabilities. Let's calculate the products!
Number of Students Running, X
P(X)
X * P(X)
1
0.05
1*0.05=0.05
2
0.15
2*0.15=0.3
3
0.1
3*0.1=0.3
4
0.15
4* 0.15=0.6
5
0.35
5 * 0.35=1.75
6
0.2
6* 0.2=1.2
To calculate the expected number of students running for president, we will sum of the values from the last column.
E(X)=0.05+0.3+0.3+ +0.6+1.75+1.2=4.2
The expected number of students that will run for president of the class is 4.2. This means that on average around 4 students will run for president.
b
To construct a relative-frequency table for 50 trials, we need to simulate numbers of students running for president. We can do that by generating random integers from 1-100, and the outcomes will represent the numbers of students running. Let's set up a proportion based on the theoretical probabilities.
Number of Students Running, X
P(X)
Represented by
1
0.05
5 numbers
2
0.15
15 numbers
3
0.10
10 numbers
4
0.15
15 numbers
5
0.35
35 numbers
6
0.20
20 numbers
Now we can choose exactly which integers from the interval 1-100 will represent which numbers of students. We can pick whichever integers we want to represent the numbers of students running, as long as the proportion is correct.
Number of Students Running, X
P(X)
Represented by
1
0.05
1-5
2
0.15
6-20
3
0.10
21-30
4
0.15
31-45
5
0.35
46-80
6
0.20
81-100
We need to make fifty trials, so let's use a graphing calculator to generate integers from 1 to 100 fifty times. Start by pushing the MATH button. Then scroll right to the PRB menu and choose the randInt( option.
Since we want to generate integers from 1 to 100 fifty times, we type 1, 100, and 50 on a graphing calculator, respectively. Then press ENTER to get fifty outcomes.
Let's say that we generated 39. This integer is in the interval 31-45, which represents the situation where 4 students run for president. Therefore, we add one to the frequency of this event and repeat for all the other randomly generated integers. Let's summarize the example results in a table.
Number of Students Running
Frequency
1
2
2
4
3
6
4
4
5
25
6
9
Note that this is a random process, so your result might be different. Now, let's calculate the relative frequencies by dividing the frequencies by the total number of trials, 50.
Number of Students Running
Frequency
Relative Frequency
1
2
2/50=0.04
2
4
4/50=0.08
3
6
6/50=0.12
4
4
4/50=0.08
5
25
25/50=0.50
6
9
9/50=0.18
This is only an example of a relative frequency table. Again, creating this table is a process affected by randomness, so your table will most likely differ.
c
For this part we will use the relative-frequency table we made in Part B. We only need to look at the number of students running and their relative frequencies.
Number of Students Running
Relative Frequency
1
0.04
2
0.08
3
0.12
4
0.08
5
0.50
6
0.18
Now we can draw an experimental probability distribution graph. To do that, we will assign a bar to each sum. The heights of the bars should be proportional to the relative frequencies of the corresponding numbers of students running for president.
The obtained graph is based on the relative-frequency table we made in Part B. Since the process of creating the table was affected by randomness, the graph is also affected by randomness. Therefore, your graph may look different.
